Soliton Velocity Research Articles

Bifurcation, bilinear forms, conservation laws and soliton solutions of the temporal-second-order KdV equation

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities [Formula: see text] and [Formula: see text] and the parameters of nonlinearity [Formula: see text] and [Formula: see text] on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

Quasi-monochromatic dynamical system of cubic–quartic optical solitons with Kerr law of nonlinear refractive index

This paper displays the quasi-monochromatic dynamical system of the cubic–quartic soliton parameters that are recovered from the conservation laws of the model. The adiabatic change in soliton velocity in presence of perturbation terms is also presented.

Detailed analysis for chirped pulses to cubic-quintic nonlinear non-paraxial pulse propagation model

This paper's main focus is on chirped pulses (CP) for a cubic-quintic nonlinear non-paraxial pulse propagation (CQ-NNP-PP) model. Chirped solitons are a relatively new single wave phenomena. The exact CPs generate from the derivative nonlinear Schrödinger equations (NLSE). Chirp is a signal with a changing frequency over time. CPs are used in spread spectrum communications as well as some sonar and radar devices. The propagation of CPs in fibre optics is getting popular due to a wide range of applications in amplification and pulse compression. In an NLSE, the dispersion management (DM) term can influence the velocity of chirp-free nonautonomous soliton but has no effect on its shape. When there is no gain, the classical optical soliton can be expressed with a variable dispersion term and nonlinearity. DM can impact the shape and motion of non autonomous solitons for CPs. We obtain hyperbolic and periodic solutions, as well as a class of solitary wave (SW) solutions such as bright, dark, singular and bell soliton solutions. The governing model will be analyzed with the aid of Jacobian elliptic functions (JEF). We also show accomplished results in 3D and 2D structures.

Nonlinear Thouless Pumping: Solitons and Transport Breakdown.

One-dimensional topological pumping of matter waves in two overlaid optical lattices moving with respect to each other is considered in the presence of attractive nonlinearity. It is shown that there exists a threshold nonlinearity level above which the matter transfer is completely arrested. Below this threshold, the transfer of both dispersive wave packets and solitons occurs in accordance with the predictions of the linear theory; i.e., it is quantized and determined by the linear dynamical Chern numbers of the lowest bands. The breakdown of the transport is also explained by nontrivial topology of the bands. In that case, the nonlinearity induces Rabi oscillations of atoms between two (or more) lowest bands. If the sum of the dynamical Chern numbers of the populated bands is zero, the oscillatory dynamics of a matter soliton in space occurs, which corresponds to the transport breakdown. Otherwise, the sum of the Chern numbers of the nonlinearity-excited bands determines the direction and magnitude of the average velocity of matter solitons that remain quantized and admit fractional values. Thus, even in the strongly nonlinear regime the topology of the linear bands is responsible for the evolution of solitons. The transition between different dynamical regimes is accurately described by the perturbation theory for solitons.

Path Integral Estimates of the Quantum Fluctuations of the Relative Soliton-Soliton Velocity in a Gross-Pitaevskii Breather

In this paper, the quantum fluctuations of the relative velocity of constituent solitons in a Gross-Pitaevskii breather are studied. The breather is confined in a weak harmonic trap. These fluctuations are monitored, indirectly, using a two-body correlation function measured at a quarter of the harmonic period after the breather creation. The results of an ab initio quantum Monte Carlo calculation, based on the Feynman-Kac path integration method, are compared with the analytical predictions using the recently suggested approach within the Bogoliubov approximation, and a good agreement is obtained.

Localized waves and breather-to-soliton conversions of the coupled Fokas–Lenells system

In this paper, we investigate the nonlinear localized waves on the plane wave backgrounds of the coupled Fokas–Lenells system, which models the ultrashort optical pulses in a birefringent optical fiber. By a given Darboux transformation, we obtain some nonlinear localized waves on the plane wave backgrounds, and provide the conversion conditions between the vector breathers and solitons. Velocities of the conversion solitons are only related to the dispersion condition and perturbation parameter. We exhibit the interaction mechanism, which results in the shape changed of the conversion solitons. The interactions of the vector breathers and rogue waves are presented through the semi-rational solutions.

Formation of microwave frequency-chirped solitons of self-induced transparency under conditions of cyclotron resonance absorption.

We study the formation of solitons of microwave self-induced transparency (M/W-SIT) which occurs under cyclotron resonance interaction of an electromagnetic pulse with an initially rectilinear magnetized electron beam. Taking into account the relativistic dependence of the gyrofrequency on the particle energy for electromagnetic wave propagating with a phase velocity different from the speed of light (i.e., far from the autoresonance conditions), such a beam can be considered as a medium of nonisochronous unexcited oscillators. Thus, similar to passing light pulses in the two-level medium, for sufficiently large amplitude and duration the incident electromagnetic pulse decomposes into one or several solitons. We find analytically the generalized solution for the M/W-SIT soliton with amplitude and duration determined, besides the soliton velocity, by the frequency self-shift parameter. The feasibility and stability of the obtained solutions are confirmed in numerical simulations of a semibounded problem describing propagation and nonlinear interaction of an incident electromagnetic pulse.

Quantum-circuit black hole lasers

A black hole laser in analogues of gravity amplifies Hawking radiation, which is unlikely to be measured in real black holes, and makes it observable. There have been proposals to realize such black hole lasers in various systems. However, no progress has been made in electric circuits for a long time, despite their many advantages such as high-precision electromagnetic wave detection. Here we propose a black hole laser in Josephson transmission lines incorporating metamaterial elements capable of producing Hawking-pair propagation modes and a Kerr nonlinearity due to the Josephson nonlinear inductance. A single dark soliton obeying the nonlinear Schrödinger equation produces a black hole-white hole horizon pair that acts as a laser cavity through a change in the refractive index due to the Kerr effect. We show that the resulting laser is a squeezed-state laser characterized by squeezing parameters. We also evaluate the degree of quantum correlation between Hawking and its partner radiations using entanglement entropy which does not require simultaneous measurements between them. As a result, the obtained entanglement entropy depending on the soliton velocity provides strong evidence that the resulting laser is derived from Hawking radiation with quantum correlation generated by pair production from the vacuum.

Analogue Hawking Radiation in Nonlinear LC Transmission Lines

Analogue systems are used to test Hawking radiation, which is hard to observe in actual black holes. One such system is the electrical transmission line, but it suffers the inevitable issue of excess heat that collapses the successfully generated analogue black holes. Soliton provides a possible solution to this problem due to its stable propagation without unnecessary energy dissipation in nonlinear transmission lines. In this work, we propose analogue Hawking radiation in a nonlinear LC transmission line including nonlinear capacitors with a third-order nonlinearity in voltage. We show that this line supports voltage soliton that obeys the nonlinear Schrödinger equation by using the discrete reductive perturbation method. The voltage soliton spatially modifies the velocity of the electromagnetic wave through the Kerr effect, resulting in an event horizon where the velocity of the electromagnetic wave is equal to the soliton velocity. Therefore, Hawking radiation bears soliton characteristics, which significantly contribute to distinguishing it from other radiation.

Nonlinear shallow water dynamics with odd viscosity

In this letter, we derive the Korteweg-de Vries (KdV) equation corresponding to the surface dynamics of a shallow depth ($h$) two-dimensional fluid with odd viscosity ($\nu_o$) subject to gravity ($g$) in the long wavelength weakly nonlinear limit. In the long wavelength limit, the odd viscosity term plays the role of surface tension albeit with opposite signs for the right and left movers. We show that there exists two regimes with a sharp transition point within the applicability of the KdV dynamics, which we refer to as weak $(|\nu_o|< \sqrt{gh^3}/6)$ and strong $(|\nu_o|> \sqrt{gh^3}/6)$ parity-breaking regimes. While the `weak' parity breaking regime results in minor qualitative differences in the soliton amplitude and velocity between the right and left movers, the `strong' parity breaking regime on the contrary results in solitons of depression (negative amplitude) in one of the chiral sectors.

Exact sub and supersonic pressure wave-fronts in nonlinear thermofluid medium

Exact sub and supersonic soliton solutions of the one-dimensional Westervelt equation are presented, describing the propagation of pressure waves with high amplitude or frequency in nonlinear acoustic media. The sub and super-sonic waves only exist on a pedestal, in the absence of which the excitations propagate at the same speed as the small amplitude waves. For nonzero background, the soliton velocity is dependent on the linear acoustic properties such as, ambient density and sound speed, as also the nonlinearity coefficient. The nonlinearity coefficient is known to vary significantly with pathological conditions of tissue, often modeled by Westervelt equation, and hence these nonlinear characteristic excitations may find application in tissue tomography.

Analytical solutions for the coupled Hirota equations in the firebringent fiber

Under investigation in this paper are the coupled Hirota (CH) equations, which describe the collision of two waves in the deep ocean and the propagation of the ultrashort optical pulses in a birefringent fiber. Based on the bilinear method, multi-soliton solutions for the CH equations are given. From the perspective of analysis, the interaction dynamics of solitons are obtained. The head-on and overtaking interactions of two/three solitons are analyzed. The elastic and inelastic interactions of two/three solitons are presented. The soliton velocity can be controlled by adjusting the physical parameters ϕj,(j=1,2,…N) and ϵ. The energy exchange occurs between the variables u and v before and after the collision. The local interference of two/three solitons is observed. The closer to the center of the interaction, the more significant the interference. The real and imaginary parts of the parameters ϕj,(j=1,2,…N) have the effects on the numbers of peaks and holes in the collisions. The structure of four eyes and one peak is observed during the two-soliton interaction. The three-soliton interactions are given with two peaks and local interference. It is hoped that the results of this study can provide some reference for the wave interaction in deep ocean, pulse propagation in optical fiber, financial option pricing, valuation of intangible assets in sports and so on.

Bilinear forms and vector bright solitons for a coupled nonlinear Schrödinger system with variable coefficients in an inhomogeneous optical fiber

In this paper, outcomes of the study on the bilinear forms and vector bright solitons for a coupled nonlinear Schrödinger system with variable coefficients are presented, which describes the simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber, where M is a positive integer. Under the integrable condition, we derive the bilinear forms, which are different from those in the existing literatures, and obtain the bright N-soliton solutions, where N is a positive integer. When M = 3, via the asymptotic analysis, we find that the amplitudes of the bright solitons are dependent on the amplification/absorption effect, , and the velocities of the bright solitons are related to the group velocity dispersion, , where z represents the spatial coordinate. Head-on, overtaking and inelastic interactions, and bound states between the two solitons are presented. We extend our analysis to M fields to obtain the vector bright N-soliton solutions.

Bilinear Bäcklund transformation, soliton and breather solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

A (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynam-ics and plasma physics is hereby investigated. Via the Hirota method, bilinear Bäcklund transformation are obtained, along with two types of the analytic solutions. Kink-shaped soliton solutions are derived via the Hirota method. Breather solutions are derived via the extended homoclinic test approach and lump solutions are obtained from the breather solutions under a limiting procedure. We find that the shape and amplitude of the one-kink soliton keep unchanged during the propagation and the velocity of the one-kink soliton depends on all the coefficients in the equation. We graphically demonstrate that the interaction between the two-kink solitons is elastic, and analyse the solitons with the influence of the coefficients. We observe that the amplitudes and shapes of the breather and lump remain unchanged during the propagation, and graphically present the breathers and lumps with the influence of the coefficients in the equation.

Dispersion and phase managed optical soliton solutions of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger equation

The complex amplitude ansatz method and He's semi-inverse method have been used to investigate the optical soliton solutions of a nonautonomous (3+1)-dimensional coupled non-linear Schrödinger (NLS) equation. The effect of the time-dependent dispersion (TD) and the phase modulation (PM) coefficients in the equation has been analyzed. The intensity profiles of the soliton solutions have been obtained by assigning different test functions to the coefficients and distinct values to the parameters, which help to understand the physical characteristics of the solutions. The intensities of two circularly polarized waves have bright, dark, parabolic, dark-bright, parabolic-bright, bright-singular, homoclinic and breather solitons that depend significantly on the dispersion and phase modulation. Some of the obtained soliton solutions are novel and did not exist in the earlier studies. It is found that the optical soliton velocity and frequency are dependent on wave numbers and dispersion of the medium but are free from the phase modulation. Moreover, the amplitude of the solitons are managed by dispersion and phase modulation both but the soliton width remains constant. The results in this study may be used for experimental investigation of transmission of dispersion and phase managed optical soliton in non-linear optical fiber.

Stability of moving Bragg solitons in a semilinear coupled system with cubic–quintic nonlinearity

We investigate the photonic bandgap structure, soliton properties and their stability in a semilinear coupled system where one core has cubic–quintic nonlinearity and is equipped with a Bragg grating, while the other one is uniform and linear. The investigation reveals that three distinct bandgaps, namely the upper, lower and central gaps, exist in the model. It is shown that the model supports two disjoint soliton families which are designated as Type 1 and Type 2 solitons. The soliton families exist only in the upper and lower bandgaps and are separated by a boundary at which no soliton solutions exist. A systematic numerical stability analysis is performed. It is found that vast regions in both the upper and lower bandgaps exist where Type 1 solitons are stable. However, Type 2 solitons are found to be always unstable. The effects of system parameters such as group velocity mismatch between the cores, velocity of solitons, and the coupling coefficient on the stability of solitons are analysed.

Effect of non-thermal electrons on ion-acoustic dressed solitons in unmagnetised plasmas

Nonlinear propagation of an ion-acoustic dressed soliton is studied in unmagnetised plasma consisting of ion, positron and non-thermal electron. The exact soliton solution is derived using the reductive perturbation method (RPM) with the help of renormalisation procedure. This exact solution reduces to the dressed soliton solution when Mach number is expanded in terms of soliton velocity. The effects of non-thermal electrons, soliton velocity and Mach number on the characteristics (amplitude, width and product of amplitude and square of width) of the KdV soliton, core structure, dressed soliton and exact soliton are discussed in detail.

Universal reductions and solitary waves of weakly nonlocal defocusing nonlinear Schrödinger equations

We study asymptotic reductions and solitary waves of a weakly nonlocal defocusing nonlinear Schrödinger (NLS) model. The hydrodynamic form of the latter is analyzed by means of multiscale expansion methods. To the leading-order of approximation (where only the first of the moments of the response function is present), we show that solitary waves, in the form of dark solitons, are governed by an effective Boussinesq/Benney–Luke (BBL) equation, which describes bidirectional waves in shallow water. Then, for long times, we reduce the BBL equation to a pair of Korteweg–de Vries (KdV) equations for right- and left-going waves, and show that the BBL solitary wave transforms into a KdV soliton. In addition, to the next order of approximation (where both the first and second moment of the response function are present), we find that dark solitons are governed by a higher-order perturbed KdV (pKdV) equation, which has been used to describe ion-acoustic solitons in plasmas and water waves in the presence of higher-order effects. The pKdV equation is approximated by a higher-order integrable system and, as a result, only insubstantial changes in the soliton shape and velocity are found, while no radiation tails (in this effective KdV picture) are produced.

Nonlinear dust acoustic waves in an inhomogeneous magnetized quantum dusty plasma

Propagations of nonlinear quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma model which is three-dimensional collisionless, and having number density gradients of its components along x-direction is investigated. This model consisting of ions, electrons and charged dust particles in the presence of an external static magnetic field. The reductive perturbation method was adopted to derive the modified Zakhavov Kusnetsov equation (MZK) whose coefficients are space dependent. The incorporation of inhomogeneity caused by nonuniform equilibrium values of particle density leads to a significant modification to the nature of nonlinear QDAWs. Solutions for the MZK equation are obtained by using a generalized expansion method. The solutions have two parts; amplitude factor and the superposition of bell-shaped solitary wave. The soliton amplitude, width, and velocity depends on the plasma inhomogeneities. Findings in these numerical investigations should be useful in understanding the astrophysical plasmas and ultra small micro- and nano-electronic devices.

Oscillatory motion of a counterpropagating Kerr soliton dimer

Both the group velocity and phase velocity of two solitons can be synchronized by a Kerr-effect mediated interaction, causing what is known as soliton trapping. Trapping can occur when solitons travel through single-pass optical fibers or when circulating in optical resonators. Here, we demonstrate and theoretically explain a new manifestation of soliton trapping that occurs between counter-propagating solitons in microresonators. When counter-pumping a microresonator using slightly detuned pump frequencies and in the presence of backscattering, the group velocities of clockwise and counter-clockwise solitons undergo periodic modulation instead of being locked to a constant velocity. Upon emission from the microcavity, the solitons feature a relative oscillatory motion having an amplitude that can be larger than the soliton pulse width. This relative motion introduces a sideband fine structure into the optical spectrum of the counter-propagating solitons. Our results highlight the significance of coherent pumping in determining soliton dynamics within microresonators and add a new dimension to the physics of soliton trapping.